Question

# The points of contact of the vertical tangents to are

- (2,5), (2,1)
- (-1,3), (5,3)
- (2,5)(5,3)
- (-1,3)(2,1)

Hint:

### A normal in geometry is a piece of geometry that is perpendicular to another piece of geometry, such as a line, ray, or vector. For instance, the (infinite) line perpendicular to the tangent line to the curve at the point is the normal line to a plane curve at the point. Here we have to find the points of contact of the vertical tangents to .

## The correct answer is: (-1,3), (5,3)

### At a specific point on a curve, tangents and normals behave just like any other straight lines. The normal runs perpendicular to the curve, and a tangent is parallel to the curve at the point. Like any other straight line, the equation of tangent and normal can be evaluated.

Here we have given equations as .

So here we used the concept of tangents and normals, we found the slope and then went to proceed for for the final answer. Using the vertical tangents we found equation and then the points.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>o</mi><mo> </mo><mi>a</mi><mi>s</mi><mo> </mo><mi>p</mi><mi>e</mi><mi>r</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>s</mi><mo> </mo><mi>t</mi><mi>h</mi><mi>e</mi><mo> </mo><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo> </mo><mi>a</mi><mi>r</mi><mi>e</mi><mo> </mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo> </mo><mi>a</mi><mi>n</mi><mi>d</mi><mo> </mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>.</mo></math>