SATHEE: Straight Line and Pair of Straight Lines 1 Question 49

Straight Line and Pair of Straight Lines 1 Question 49

49. No tangent can be drawn from the point $(5 / 2, 1)$ to the circumcircle of the triangle with vertices $(1, \sqrt{3})$ , $(1, - \sqrt{3})$ and $(3, \sqrt{3})$ .

(1985, 1M)

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Solution:

Since, $(1, \sqrt{3}), (1, - \sqrt{3})$ and $(3, \sqrt{3})$ form a right angled triangle at $(1, \sqrt{3})$

$∴$ Equation of circumcircle taking $(3, \sqrt{3})$ and $(1, - \sqrt{3})$ as end points of diameter.

$\begin{aligned} ∴ & (x - 3) (x - 1) + (y - \sqrt{3}) (y + \sqrt{3}) & = 0 \\ \Rightarrow & x^{2} - 4 x + 3 + y^{2} - 3 & = 0 \\ \Rightarrow & x^{2} + y^{2} - 4 x & = 0 \end{aligned}$

At point

$\frac{5}{2}, 1, S_{1} = \frac{25}{4} + 1 - 10 < 0$

$∴$ Point $(5 / 2, 1)$ lies inside the circle.

Hence, no tangent can be drawn.

Hence, given statement is true.

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Straight Line and Pair of Straight Lines 1 Question 49

49. No tangent can be drawn from the point (5/2,1) to the circumcircle of the triangle with vertices (1,3), (1,−3) and (3,3).

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49. No tangent can be drawn from the point $(5 / 2, 1)$ to the circumcircle of the triangle with vertices $(1, \sqrt{3})$ , $(1, - \sqrt{3})$ and $(3, \sqrt{3})$ .