Question
In the xy-plane, a line that has the equation y = c for some constant c intersects a parabola at exactly one point. If the parabola has the equation
, what is the value of c ?
Hint:
Hint:
- When a quadratic equation has same solution then its Discriminant will be equal to zero.
The correct answer is: 6.25
Explanation:
- We have given a parabola with equation
and a line y = c
- We have to find the value of c.
Step 1 of 1:
The given equation of parabola is
and a line y = c
If they intersect at only one point. Then,
has only one solution. Or same solution.
So,
D = 0
![left parenthesis 5 right parenthesis squared minus 4 left parenthesis negative 1 right parenthesis left parenthesis negative c right parenthesis equals 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAAARCAYAAADKSt4yAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAPUJuPDwAAAtxJREFUeNrtWT1oFEEUXsIRLA7BMoUIEsRCQiBYiIjYhCBiEbCwDgQJRxCxE0vhsBILQUQs7CSkEhEkiBxB0oRgISKktAiChYUcy0J8o5+yDu/NzO7Om9vD++BrZu92vvez++a9zTIdHII5cZs4m00wgYU1Yr/C7/v4j4QpXN8dQ9tiwOef1Jp8epJhjvi+xv+28V8XhmNqWwxI/hmVppB4JRExz5RCjmUsEAeO+14kvkug/wqjTbItJk4R7xL3hOuSfzQ1ueCLVxMcJT4i7oCPsfYP5mG8fe4KxQBG2JjBfU8qO7BL/Cho5myLiefEVY+/bP9oa6obr6bYIl61XgBb9o/uE3sNkm2dOXuYJ/4lEk4b5glaETRztmk1RaH+SaWpSrya4hrxIbP+gHi9vPCaeKFBsp2zMtgk2CvuFaqA86W9Oc2cbamTzfaPtqYuEvoLJgKbxGMOPTGwQVxk1i/h2l98J043SLZp3OMPTKKdThBgs+8H4gmHZs621Mlm+0dTUxfd/z087Gaf2yhpkp4s8Kx+6LDzm2CTWTsoLxTCpjlEmeS5BUMk5B7BWq18zxPwIlFp8tmYJ9LUF8qZS08MFKF7uX7YwWHSdFyfHG+svEGQ6jxJc4FNTaG0v2ay1dVk5ppfrZLZhmQb1ik1i8IYw/Va1oJprWcDAv4/ldGzgWMNjTJ6INh0xC6jb3DQDsEw4ACcqmyFOKSKbakaBC1Nl9EMVG3oYjUIS8ILaqPOeOAMcZ9Z7+Eeo0ZbRx+2f7Q0LeCo44NGvJaJT5j1Z/bogxsybuIJmAKXkGjLzA21hoQxAp5qgNqWoe4eOtEOutGbzFtUK15vsV8HJfWOcOz69Z2u/M3MDOk+4+Bn2toXOBNwT9MgawekgNu2aZfzEP9oaTqOABcYC60kjJdpTJ7iqPUj+z1s70rdndaH+FFj8iG+hfG6kVX7jGHq/mo2HqhqWwz4/JNa00jj9RP1gweqNspITAAAAQF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+KDwvbW8+PG1uPjU8L21uPjxtc3VwPjxtbz4pPC9tbz48bW4+MjwvbW4+PC9tc3VwPjxtbz4mI3gyMjEyOzwvbW8+PG1uPjQ8L21uPjxtbz4oPC9tbz48bW8+JiN4MjIxMjs8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PG1vPig8L21vPjxtbz4mI3gyMjEyOzwvbW8+PG1pPmM8L21pPjxtbz4pPC9tbz48bW8+PTwvbW8+PG1uPjA8L21uPjwvbWF0aD4UsqrQAAAAAElFTkSuQmCC)
25 - 4c = 0
c = 6.25
Hence, The value of c = 6.25 .
Related Questions to study
![table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 2.4 x minus 1.5 y equals 0.3 end cell row cell 1.6 x plus 0.5 y equals negative 1.3 end cell end table](data:image/png;base64,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)
The system of equations above is graphed in the xy -plane. What is the x -coordinate of the intersection point ( x, y) of the system?
![table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 2.4 x minus 1.5 y equals 0.3 end cell row cell 1.6 x plus 0.5 y equals negative 1.3 end cell end table](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAnCAYAAADpYmEPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAABFlJREFUeNrtXG9kVWEYP+a6JonMTPowJpnMRJLMTMxMMjMmSZLRh+nD9DXpQyJ9mEliMskkck0mE5mZScZMkiTSpz7sSx8mc11jPY9+V8fxvve877nvOe+7eX782D33nPP+eZ73eZ733PNbFOWHPmKFuE2sET8Rr1re4xJxLwoHJ4l3MZY07DWgLxwhPiWug7M4psMF4hJxh1glfifOENt8dH6VeIV4GJ9PET/gmAn4uq+BOdQ88aZhn0Lqdx3LxJHEgl1ucP4KcYxYjh07TXwXyoA6iZ8Nz+XVMxGoYfajQ40THyuOz1gs8jq2QxpY1TBdLqcYhp3toeL4fXy3XxxqjnhZcXyK+MBhn7n8GNKktYrFfbqI30JxpvNIe41QRhTrNDDMJrEj9vkG8ck+i1DXiY8Sxw6hXmmzrM0a1Wi/E6krPt9bBv3swPxyvy6G4EytKAT7Us7jqHPL0DDDxGn8PZhSD/hwqBrSAxe2t2P1ZBxsnNeJY/dQ/LvEboPvahabi6kQnOko8Y0m5MbRq4hgacZ7TxzAzqvNwMhZVncz9VGJeAYOwqmiO/E99/lL7HM78QcWYFEOVTW04SiCwoBPZ+qCM50wOHddcV6a8aawwnoDK8pVGMLuN4md2N/TKVEg66LY0qS8VsOUF999r/tyJl6Nz1ATmBrKZqLqz7qmM+xUfDiULhqsYeF1ITq15NDnCkoElZNXHIwhd3SgNijlZDye/EU4K6+ajai4B25ZHaoHDpPECzwfekm8llOfx7C4k3huuRh7UZgXjreKeqFRVLIxXjsK3bgD8UO6ec8OFR/LAna1LeAwnGlMk7bnIvNndFmxgrZKSH93FCk4PoZVPL8qYQz8iOEndqaFwzR12TpUGcY6rjj3lSas5zkm3VjGsZJ3sWXnaH1Wc99RXDdSwOZoDimL67ZZxc4zPoZ+BIYquIKFKwgc7EgfZRoErrCE9CgQNI0epBWBQCAQCAQCgUAgEAgEAoFAIBCECBsdWh38Ayj/EMqvvvIPivyKyYDncdjq0ULU1PmwqXNdno0OLcK5i+h03ZAs7lzzPIm2erS9AxwkbGyamy7PpHEWcm4UODGmRs+iRzvIDuVijNtFND4bmb/550JTZzohWfRooWnqQnIoJ7o8k8a5kWMW92xWU2c6IVn0aKFp6kJwKKe6PJPGq6idFlDE1ZACdf8go1lNnemEZNGjhaap8+lQuejyTIWNm5jk+jvIffDoSc01RWjqmtGjhaKpyyPK2UYop7o8k8Z5Nave8+ZdwS/NNc1o6kwnxJUezaemLqQayokuz6TxpUivJ1OllmY1dTZFeZ56tCI0daHt8qpFND6p2fF0KzzahabOdEJc6dF8aupCcignujwTHVoZKWEi+i/uPBf905oNJuoMF5o6mwmx1aOFqKkryqFy1eXZauraEQ3+oBjmDvUntuo+NHW2erQQNXV5FvW6eRBdngeIpk7gFKKpEziDaOoS+AsHysGQVTkM4gAAAbh0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXRhYmxlIGNvbHVtbmFsaWduPSJyaWdodCBsZWZ0IHJpZ2h0IGxlZnQgcmlnaHQgbGVmdCByaWdodCBsZWZ0IHJpZ2h0IGxlZnQgcmlnaHQgbGVmdCIgY29sdW1uc3BhY2luZz0iMGVtIDJlbSAwZW0gMmVtIDBlbSAyZW0gMGVtIDJlbSAwZW0gMmVtIDBlbSI+PG10cj48bXRkPjxtbj4yLjQ8L21uPjxtaT54PC9taT48bW8+JiN4MjIxMjs8L21vPjxtbj4xLjU8L21uPjxtaT55PC9taT48bW8+PTwvbW8+PG1uPjAuMzwvbW4+PC9tdGQ+PC9tdHI+PG10cj48bXRkPjxtbj4xLjY8L21uPjxtaT54PC9taT48bW8+KzwvbW8+PG1uPjAuNTwvbW4+PG1pPnk8L21pPjxtbz49PC9tbz48bW8+JiN4MjIxMjs8L21vPjxtbj4xLjM8L21uPjwvbXRkPjwvbXRyPjwvbXRhYmxlPjwvbWF0aD4KKEdDAAAAAElFTkSuQmCC)
The system of equations above is graphed in the xy -plane. What is the x -coordinate of the intersection point ( x, y) of the system?
![x minus 2 y equals negative 3](data:image/png;base64,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)
![x plus y equals 21](data:image/png;base64,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)
According to the system of equations above, what is the value of X ?
Note:
Here we find the value of y from equation (1) and use it in equation (2).
We could do it the other way and receive the same answer, that is, if we find the value of y from equation (2) and use it in equation (1) to find x, we get the same value of x as found in the solution above.
Students are encouraged to try this method too.
![x minus 2 y equals negative 3](data:image/png;base64,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)
![x plus y equals 21](data:image/png;base64,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)
According to the system of equations above, what is the value of X ?
Note:
Here we find the value of y from equation (1) and use it in equation (2).
We could do it the other way and receive the same answer, that is, if we find the value of y from equation (2) and use it in equation (1) to find x, we get the same value of x as found in the solution above.
Students are encouraged to try this method too.