CBSE Class 11-science Answered
Three points P(h,k), Q (x,y) and R (x,y) lie on a line show that (h-x) (y-y)= (k-y) (x-x),
Asked by hrai7387 | 16 Jan, 2023, 19:36: PM
![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/17a6d651076dee5c5f68e3b66d8a393963c566c6e6e8f8.01921892f5.png)
Let (h.k) , (x1 , y1) and (x2 , y2 ) are three given points
slope of segment PQ = ![begin mathsize 14px style fraction numerator left parenthesis y subscript 1 minus k right parenthesis over denominator left parenthesis x subscript 1 minus h right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/1d40f8c237a630f3ef3f1f0ec13fcf87.png)
![begin mathsize 14px style fraction numerator left parenthesis y subscript 1 minus k right parenthesis over denominator left parenthesis x subscript 1 minus h right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/1d40f8c237a630f3ef3f1f0ec13fcf87.png)
slope of segment QR = ![begin mathsize 14px style fraction numerator left parenthesis y subscript 2 minus y subscript 1 right parenthesis over denominator left parenthesis x subscript 2 minus x subscript 1 right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/68f23eb0f2ca5484a1f3f5cec4ce36dc.png)
![begin mathsize 14px style fraction numerator left parenthesis y subscript 2 minus y subscript 1 right parenthesis over denominator left parenthesis x subscript 2 minus x subscript 1 right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/68f23eb0f2ca5484a1f3f5cec4ce36dc.png)
If P, Q and R on same line , then both the above slopes are equal and we have
![begin mathsize 14px style fraction numerator left parenthesis y subscript 1 minus k right parenthesis over denominator left parenthesis x subscript 1 minus h right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/1d40f8c237a630f3ef3f1f0ec13fcf87.png)
![begin mathsize 14px style fraction numerator left parenthesis y subscript 2 minus y subscript 1 right parenthesis over denominator left parenthesis x subscript 2 minus x subscript 1 right parenthesis end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/68f23eb0f2ca5484a1f3f5cec4ce36dc.png)
By cross multiplication of above expression, we get
![begin mathsize 14px style left parenthesis y subscript 1 space minus space k space right parenthesis space left parenthesis space x subscript 2 minus x subscript 1 right parenthesis space equals space left parenthesis x subscript 1 minus h right parenthesis left parenthesis y subscript 2 minus y subscript 1 right parenthesis end style](https://images.topperlearning.com/topper/tinymce/cache/4281b731123252bb557a8e89be947251.png)
Answered by Thiyagarajan K | 16 Jan, 2023, 20:39: PM
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