Question

The tangent at (3,4) and (4,-3) to the circle $x^{2}+y^{2}=25$ are

A. coincident
B. parallel
C. perpendicular
D. at an angle of $45^{\circ}$

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Answer

To determine the relationship between the tangents at the points (3, 4) and (4, -3) on the circle $x^2 + y^2 = 25$, follow these steps:

  1. Equation of the Circle: The given circle is $x^2 + y^2 = 25$.

  2. Points of Tangency: The points of tangency are (3, 4) and (4, -3).

  3. Equation of Tangent to a Circle: The standard form of the equation of a tangent to a circle $x^2 + y^2 = r^2$ at a point $(x_1, y_1)$ on the circle is: $$ x_1 x + y_1 y = r^2 $$

  4. Tangents Equations:

    • For the point (3, 4): $$ 3x + 4y = 25 $$
    • For the point (4, -3): $$ 4x - 3y = 25 $$
  5. Finding Slopes:

    • The slope ($m_1$) of the tangent $3x + 4y = 25$: $$ m_1 = -\frac{a}{b} = -\frac{3}{4} $$
    • The slope ($m_2$) of the tangent $4x - 3y = 25$: $$ m_2 = -\frac{a}{b} = -\frac{4}{-3} = \frac{4}{3} $$
  6. Checking Perpendicularity: The product of the slopes of two perpendicular lines is $-1$. So, check: $$ m_1 \cdot m_2 = \left(-\frac{3}{4}\right) \cdot \left(\frac{4}{3}\right) = -1 $$ Since the product of the slopes is $-1$, the tangents are perpendicular.

Thus, the tangents at the points (3, 4) and (4, -3) to the circle $x^2 + y^2 = 25$ are perpendicular.

Answer: C. Perpendicular


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